Question:
Let a1 = 1, b1 = 2 and c1 = 3. Consider the convergent sequences
\(\left\{a_n\right\}^{\infin}_{n=1},\left\{b_n\right\}^{\infin}_{n=1} \text{ and }\left\{c_n\right\}^{\infin}_{n=1}\)
defined as follows :
\(a_{n+1}=\frac{a_n+b_n}{2},b_{n+1}=\frac{b_n+c_n}{2} \text{ and } c_{n+1}=\frac{c_n+a_n}{2} \text{ for } n \ge1.\)
Then,
\(\sum\limits_{n=1}^{\infin}b_nc_n(a_{n+1}-a_n)+\sum\limits_{n=1}^{\infin}(b_{n+1}c_{n+1}-b_nc_n)a_{n+1}\)
equals _____________ (rounded off to two decimal places)
Let a1 = 1, b1 = 2 and c1 = 3. Consider the convergent sequences
\(\left\{a_n\right\}^{\infin}_{n=1},\left\{b_n\right\}^{\infin}_{n=1} \text{ and }\left\{c_n\right\}^{\infin}_{n=1}\)
defined as follows :
\(a_{n+1}=\frac{a_n+b_n}{2},b_{n+1}=\frac{b_n+c_n}{2} \text{ and } c_{n+1}=\frac{c_n+a_n}{2} \text{ for } n \ge1.\)
Then,
\(\sum\limits_{n=1}^{\infin}b_nc_n(a_{n+1}-a_n)+\sum\limits_{n=1}^{\infin}(b_{n+1}c_{n+1}-b_nc_n)a_{n+1}\)
equals _____________ (rounded off to two decimal places)
\(\left\{a_n\right\}^{\infin}_{n=1},\left\{b_n\right\}^{\infin}_{n=1} \text{ and }\left\{c_n\right\}^{\infin}_{n=1}\)
defined as follows :
\(a_{n+1}=\frac{a_n+b_n}{2},b_{n+1}=\frac{b_n+c_n}{2} \text{ and } c_{n+1}=\frac{c_n+a_n}{2} \text{ for } n \ge1.\)
Then,
\(\sum\limits_{n=1}^{\infin}b_nc_n(a_{n+1}-a_n)+\sum\limits_{n=1}^{\infin}(b_{n+1}c_{n+1}-b_nc_n)a_{n+1}\)
equals _____________ (rounded off to two decimal places)
Updated On: Oct 1, 2024
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Correct Answer: 1.95
Solution and Explanation
The correct answer is 1.95 to 2.05. (approx)
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